Chapter 7.13, Problem 1MP

The displacement (from equilibrium) of a particle executing simple harmonic motion may be either $y=A\sin \omega t$ or $y=A\sin (\omega t+\varphi )$ depending on our choice of time origin. Show that the average of the kinetic energy of a particle of mass $m$ (over a period of the motion) is the same for the two formulas (as it must be since both describe the same physical motion). Find the average value of the kinetic energy for the $\sin (\omega t+\varphi )$ case in two ways:

(a) By selecting the integration limits (as you may by Problem 4.1) so that a change of variable reduces the integral to the sin $\omega t$ case.

(b) By expanding $\sin (\omega t+\varphi )$ by the trigonometric addition formulas and using (5.2) to write the average values.